# Quaternion

## Prompt

In this reference frame, the spaceship is oriented at `[0, 0, 1]`

relative to the spacecraft.
In the reference frame of J2000 the spaceship is oriented at `[0.14129425, -0.98905974, 0.04238827]`

.
Find the spacecraft attitude quaternion.

## Background

In mathematics, a quaternion is an extended complex number which helpes to represent mechanics, orientation, rotation, and movement in 3 dimensions.
They are often used in computer graphics, computer vision, and applied mathematics and represented in the form a + b**i** + c**j** + d**k**.
In that form a, b, c, and d are real numbers and **i**, **j**, and **k** are the basic quaternions.

### To learn more about quaternions

## Solution

Looking up how to convert two vectors into a quaternion we come across the set of equations

```
A, B are the given vectors
The form of the answer will be q = (w, x, y, z) <-- NOTE the order
```

$$N = \frac{A\times B}{||A\times B||}$$

$$\Theta= \arctan{\frac{||A\times B||}{A\cdot B}}$$

$$q = [\cos(\frac{\Theta}{2}), \vec{N}\sin(\frac{\Theta}{2})]$$

So for our solution, A is `[0, 0, 1]`

and B is `[0.14129425, -0.98905974, 0.04238827]`

.
First we find **N** and then **Theta** and then plug into the equation for **q**.
Once we do that, we input the values into the netcat connection (during the competition) and…

**QED**